The dump file is placed in that location at the next reboot. This is because when you get a BSOD, the kernel dumps memory into the pagefile. It can't create or even open any other files at this point. Next time the kernel boots, it checks existing pagefiles for a signature that indicates whether or not the pagefile contains a dump from previous session. If so, it moves the pagefile into the dump file location you have specified.

But this whole concept also means that directly after a crash, you can use another operating system to copy the pagefile from the crashed system and that file will then contain the dump. It also means that if you don't have a pagefile or if it is not big enough, you would not get any memory dumps.

There are a couple of brick walls you could hit anyway. Even if you workaround the principle that only the pagefile at system partition can be used as dump file (DedicatedDumpFile setting), certain limitations still apply. All drivers along the path from kernel though filesystem down to disk volume and physical disk need to be capable of handling dump files. It is a specific scenario in the kernel for which there needs to be specific support in related drivers. This is usually no problems for physical disk drivers of various kinds because they are always designed for this scenario anyway and they get appropriate notifications from the kernel to be able to prepare correctly for it when the pagefile is created. But if you use some kind of disk virtualization to create a pagefile where I/O requires network drivers or other kinds of drivers that normally are not related to dump files, this will likely fail.

I seem to recall that I have at some point read about someone creating a dump file on a physical disk on the machine where WinPE ran, but I cannot seem to find anything that right now. I would say that would be the only option that possibly could work, unless you find some kind of highly specialized driver for the particular purpose of creating a pagefile that can be used to store a dump at some special location later.

I personally have pretty much only used live debugging (windbg/kd/etc) to debug PE or RE sessions, not dump files. That works pretty much in the same ways as for non-PE Windows sessions. But I can of course understand that there might be scenarios where dump files would be the only practical option to investigate some problems so if you find a solution, please share it! Could be useful to know!